lmomgld {lmomco} | R Documentation

## L-moments of the Generalized Lambda Distribution

### Description

This function estimates the L-moments of the Generalized Lambda distribution given the parameters (xi, α, kappa, and h) from `vec2par`. The L-moments in terms of the parameters are complicated; however, there are analytical solutions. There are no simple expressions of the parameters in terms of the L-moments. The first L-moment or the mean of the distribution is

λ_1 = xi + α (frac{1}{kappa+1} - frac{1}{h+1} ) mbox{.}

The second L-moment or L-scale in terms of the parameters and the mean is

λ_2 = xi + frac{2α}{(kappa+2)} - 2α ( frac{1}{h+1} - frac{1}{h+2} ) - xi mbox{.}

The third L-moment in terms of the parameters, the mean, and L-scale is

mbox{boldmath $Y$} = 2xi + frac{6α}{(kappa+3)} - 3(α+xi) + xi mbox{ and}

λ_3 = mbox{boldmath $Y$} + 6α (frac{2}{h+2} - frac{1}{h+3} - frac{1}{h+1}) mbox{.}

The fourth L-moment in termes of the parameters and the first three L-moments is

mbox{boldmath $Y$} = frac{-3}{h+4}(frac{2}{h+2} - frac{1}{h+3} - frac{1}{h+1}) mbox{,}

mbox{boldmath $Z$} = frac{20xi}{4} + frac{20α}{(kappa+4)} - 20 mbox{boldmath $Y$}α mbox{, and}

λ_4 = mbox{boldmath $Z$} - 5(kappa + 3(α+xi) - xi) + 6(α + xi) - xi mbox{.}

It is conventional to express L-moments in terms of only the parameters and not the other L-moments. Lengthy algebra and further manipulation yields such a system of equations. The L-moments of the distribution are

λ_1 = xi + α (frac{1}{kappa+1} - frac{1}{h+1} ) mbox{,}

λ_2 = α (frac{kappa}{(kappa+2)(kappa+1)} + frac{h}{(h+2)(h+1)}) mbox{,}

λ_3 = α (frac{kappa (kappa - 1)} {(kappa+3)(kappa+2)(kappa+1)} - frac{h (h - 1)} {(h+3)(h+2)(h+1)} ) mbox{, and}

λ_4 = α (frac{kappa (kappa - 2)(kappa - 1)} {(kappa+4)(kappa+3)(kappa+2)(kappa+1)} + frac{h (h - 2)(h - 1)} {(h+4)(h+3)(h+2)(h+1)} ) mbox{.}

The L-moment ratios are

tau_3 = frac{kappa(kappa-1)(h+3)(h+2)(h+1) - h(h-1)(kappa+3)(kappa+2)(kappa+1)} {(kappa+3)(h+3) times [kappa(h+2)(h+1) + h(kappa+2)(kappa+1)] } mbox{ and}

tau_4 = frac{kappa(kappa-2)(kappa-1)(h+4)(h+3)(h+2)(h+1) + h(h-2)(h-1)(kappa+4)(kappa+3)(kappa+2)(kappa+1)} {(kappa+4)(h+4)(kappa+3)(h+3) times [kappa(h+2)(h+1) + h(kappa+2)(kappa+1)] } mbox{.}

The pattern being established through symmetry, even higher L-moment ratios are readily obtained. Note the alternating substraction and addition of the two terms in the numerator of the L-moment ratios (tau_r). For odd r >= 3 substraction is seen and for even r >= 3 addition is seen. For example, the fifth L-moment ratio is

N1 = kappa(kappa-3)(kappa-2)(kappa-1)(h+5)(h+4)(h+3)(h+2)(h+1) mbox{,}

N2 = h(h-3)(h-2)(h-1)(kappa+5)(kappa+4)(kappa+3)(kappa+2)(kappa+1) mbox{,}

D1 = (kappa+5)(h+5)(kappa+4)(h+4)(kappa+3)(h+3) mbox{,}

D2 = [kappa(h+2)(h+1) + h(kappa+2)(kappa+1)] mbox{, and}

tau_5 = frac{N1 - N2}{D1 times D2} mbox{.}

By inspection the tau_r equations are not applicable for negative integer values k={-1, -2, -3, -4, ... } and h={-1, -2, -3, -4, ... } as division by zero will result. There are additional, but difficult to formulate, restrictions on the parameters both to define a valid Generalized Lambda distribution as well as valid L-moments. Verification of the parameters is conducted through `are.pargld.valid`, and verification of the L-moment validity is conducted through `are.lmom.valid`.

### Usage

    lmomgld(gldpara)

### Arguments

`gldpara` | The parameters of the distribution.

### Value

An R `list` is returned.

`L1` | Arithmetic mean.
`L2` | L-scale—analogous to standard deviation.
`LCV` | coefficient of L-variation—analogous to coe. of variation.
`TAU3` | The third L-moment ratio or L-skew—analogous to skew.
`TAU4` | The fourth L-moment ratio or L-kurtosis—analogous to kurtosis.
`TAU5` | The fifth L-moment ratio.
`L3` | The third L-moment.
`L4` | The fourth L-moment.
`L5` | The fifth L-moment.
`source` | An attribute identifying the computational source of the L-moments: “lmomgld”.

### Author(s)

W.H. Asquith

### Source

Derivations conducted by W.H. Asquith on February 11 and 12, 2006.

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Karvanen, J., Eriksson, J., and Koivunen, V., 2002, Adaptive score functions for maximum likelihood ICA: Journal of VLSI Signal Processing, vol. 32, p. 82–92.

Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distibutions—The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.

### See Also

`pargld, \code{cdfgld}, \code{quagld}`

### Examples

    lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))

[Package lmomco version 0.92 Index]